Q. 67 - Calculus

Question

$$\begin{gathered} \mathrm{Recall} \mathrm{that} \mathrm{the} \mathrm{volume}, V, \mathrm{of} \mathrm{a} \mathrm{cylinder} \mathrm{is} \mathrm{given} \mathrm{by} V(r, h)=\pi r^{2}h. \mathrm{Find} \frac{\partial V}{\partial r} \mathrm{and} \frac{\partial V}{\partial h}. \\ \mathrm{What} \mathrm{are} \mathrm{the} \mathrm{physical} \mathrm{interpretations} \mathrm{of} \mathrm{these} \mathrm{partial} \mathrm{derivatives}? \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \frac{\partial V}{\partial r}=2\pi rh \\ \mathrm{This} \mathrm{tells} \mathrm{us} \mathrm{how} \mathrm{fast} \mathrm{the} \mathrm{volume} \mathrm{changes} \mathrm{with} \mathrm{a} \mathrm{unit} \mathrm{change} \mathrm{in} \mathrm{the} \mathrm{radius}. \\ \frac{\partial V}{\partial h}=\pi r^2 \\ \mathrm{This} \mathrm{tells} \mathrm{us} \mathrm{how} \mathrm{fast} \mathrm{the} \mathrm{volume} \mathrm{changes} \mathrm{with} \mathrm{a} \mathrm{unit} \mathrm{change} \mathrm{in} \mathrm{the} \mathrm{height}. \end{gathered}$$

Step 1. Given information

The volume of a cylinder is given by,$V(r, h)=\pi r^{2}h$

Step 2. Finding the partial derivatives and physical interpretation of them

$$\begin{gathered} \frac{\partial V}{\partial r}=2\pi rh \\ \mathrm{This} \mathrm{tells} \mathrm{us} \mathrm{how} \mathrm{fast} \mathrm{the} \mathrm{volume} \mathrm{changes} \mathrm{with} \mathrm{a} \mathrm{unit} \mathrm{change} \mathrm{in} \mathrm{the} \mathrm{radius}. \\ \frac{\partial V}{\partial h}=\pi r^2 \\ \mathrm{This} \mathrm{tells} \mathrm{us} \mathrm{how} \mathrm{fast} \mathrm{the} \mathrm{volume} \mathrm{changes} \mathrm{with} \mathrm{a} \mathrm{unit} \mathrm{change} \mathrm{in} \mathrm{the} \mathrm{height}. \end{gathered}$$